Let n be a positive odd integer. Prove that n4 ≡ 1 (mod 16). You may use any of the mod rules provided in class (In addition, if you end up getting an expression of the form a (3a+1) somewhere in your work where a is an integer, separately end up proving that this expression is always even and use that result to complete your proof).

If is odd, then we can write for some .

Expanding gives

This reduces to

Notice that is 8 times the product of two consecutive integers, one even and the other odd. This product is even and thus divisible by 2, so that where , and the equivalence further reduces to

as required.

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step-by-step explanation:

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